New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator

In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f ( t ) appearing inside the integral sign of the definition of the CF operator. Thus, a numerical differentiation formula has been proposed in the present study. The obtained numerical approximation was found to be of first-order convergence, having decreasing absolute errors with respect to a decrease in the time step size h used in the approximations. Such absolute errors are computed as the absolute difference between the results obtained through the proposed numerical approximation and the exact solution. With the aim of improved accuracy, the two-point finite forward difference formula has also been utilized for the continuous temporal mesh. Some mathematical models of varying nature, including a diffusion-wave equation, are numerically solved, whereas the first-order accuracy is not only verified by the error analysis but also experimentally tested by decreasing the time-step size by one order of magnitude, whereupon the proposed numerical approximation also shows a one-order decrease in the magnitude of its absolute errors computed at the final mesh point of the integration interval under consideration.

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Efficient Time-Stepping/Spectral Methods for the Navier-Stokes-Nernst-Planck-Poisson Equations

Communications in Computational Physics ◽

10.4208/cicp.191015.260816a ◽

2017 ◽

Vol 21 (5) ◽

pp. 1408-1428 ◽

Cited By ~ 1

Author(s):

Xiaoling Liu ◽

Chuanju Xu

Keyword(s):

Stability Condition ◽

Time Discretization ◽

Equation System ◽

Second Order ◽

Navier Stokes ◽

Step Size ◽

Time Step ◽

Time Step Size ◽

First Order ◽

Time Stepping

AbstractThis paper is concerned with numerical methods for the Navier-Stokes-Nernst-Planck-Poisson equation system. The main goal is to construct and analyze some stable time stepping schemes for the time discretization and use a spectral method for the spatial discretization. The main contribution of the paper includes: 1) an useful stability inequality for the weak solution is derived; 2) a first order time stepping scheme is constructed, and the non-negativity of the concentration components of the discrete solution is proved. This is an important property since the exact solution shares the same property. Moreover, the stability of the scheme is established, together with a stability condition on the time step size; 3) a modified first order scheme is proposed in order to decouple the calculation of the velocity and pressure in the fluid field. This new scheme equally preserves the non-negativity of the discrete concentration solution, and is stable under a similar stability condition; 4) a stabilization technique is introduced to make the above mentioned schemes stable without restriction condition on the time step size; 5) finally we construct a second order finite difference scheme in time and spectral discretization in space. The numerical tests carried out in the paper show that all the proposed schemes possess some desirable properties, such as conditionally/unconditionally stability, first/second order convergence, non-negativity of the discrete concentrations, and so on.

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EFFECTS OF TIME STEP SIZE IN IMPLICIT DYNAMIC ROUTING

JAWRA Journal of the American Water Resources Association ◽

10.1111/j.1752-1688.1973.tb01741.x ◽

1973 ◽

Vol 9 (2) ◽

pp. 338-351 ◽

Cited By ~ 10

Author(s):

D. L. Fread

Keyword(s):

Dynamic Routing ◽

Step Size ◽

Time Step ◽

Time Step Size

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Analysis of a Decoupled Time-Stepping Scheme for Evolutionary Micropolar Fluid Flows

Advances in Numerical Analysis ◽

10.1155/2016/7010645 ◽

2016 ◽

Vol 2016 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

S. S. Ravindran

Keyword(s):

Micropolar Fluid ◽

Stokes Equations ◽

Fluid Model ◽

Navier Stokes ◽

Optimal Order ◽

Step Size ◽

Time Step ◽

Order Error ◽

Time Step Size ◽

Time Stepping

Micropolar fluid model consists of Navier-Stokes equations and microrotational velocity equations describing the dynamics of flows in which microstructure of fluid is important. In this paper, we propose and analyze a decoupled time-stepping algorithm for the evolutionary micropolar flow. The proposed method requires solving only one uncoupled Navier-Stokes and one microrotation subphysics problem per time step. We derive optimal order error estimates in suitable norms without assuming any stability condition or time step size restriction.

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Overcoming the Barrier on Time Step Size in Multiscale Molecular Dynamics Simulation of Molecular Liquids

Journal of Chemical Theory and Computation ◽

10.1021/ct200157x ◽

2011 ◽

Vol 8 (1) ◽

pp. 6-16 ◽

Cited By ~ 10

Author(s):

Igor P. Omelyan ◽

Andriy Kovalenko

Keyword(s):

Molecular Dynamics ◽

Molecular Dynamics Simulation ◽

Dynamics Simulation ◽

Molecular Liquids ◽

Step Size ◽

Time Step ◽

Time Step Size

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On the Influence of Inflow Model Selection for Time-Domain Tiltrotor Aeroelastic Analysis

Journal of the American Helicopter Society ◽

10.4050/jahs.66.032009 ◽

2021 ◽

Author(s):

Ethan Corle ◽

Matthew Floros ◽

Sven Schmitz

Keyword(s):

Experimental Data ◽

Particle Method ◽

Comprehensive Analysis ◽

Solution Stability ◽

Step Size ◽

Time Step ◽

Time Step Size ◽

Vortex Particle Method ◽

Whirl Flutter ◽

Vortex Particle

The methods of using the viscous vortex particle method, dynamic inflow, and uniform inflow to conduct whirl-flutter stability analysis are evaluated on a four-bladed, soft-inplane tiltrotor model using the Rotorcraft Comprehensive Analysis System. For the first time, coupled transient simulations between comprehensive analysis and a vortex particle method inflow model are used to predict whirl-flutter stability. Resolution studies are performed for both spatial and temporal resolution in the transient solution. Stability in transient analysis is noted to be influenced by both. As the particle resolution is refined, a reduction in simulation time-step size must also be performed. An azimuthal time step size of 0.3 deg is used to consider a range of particle resolutions to understand the influence on whirl-flutter stability predictions. Comparisons are made between uniform inflow, dynamic inflow, and the vortex particle method with respect to prediction capabilities when compared to wing beam-bending frequency and damping experimental data. Challenges in assessing the most accurate inflow model are noted due to uncertainty in experimental data; however, a consistent trend of increasing damping with additional levels of fidelity in the inflow model is observed. Excellent correlation is observed between the dynamic inflow predictions and the vortex particle method predictions in which the wing is not part of the inflow model, indicating that the dynamic inflow model is adequate for capturing damping due to the induced velocity on the rotor disk. Additional damping is noted in the full vortex particle method model, with the wing included, which is attributed to either an interactional aerodynamic effect between the rotor and the wing or a more accurate representation of the unsteady loading on the wing due to induced velocities.

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A Strategy to Accelerate the Numerical Integration in the Dynamic Simulation of Multibody Systems

Volume 1A: 16th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc97/vib-4228 ◽

1997 ◽

Author(s):

Jesús Cardenal ◽

Javier Cuadrado ◽

Eduardo Bayo

Keyword(s):

Multibody Systems ◽

Equations Of Motion ◽

Stiff Systems ◽

Step Size ◽

Step Method ◽

Integral Of Motion ◽

Time Step ◽

Time Step Size ◽

Variable Time ◽

Numerical Integrator

Abstract This paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that becomes more accurate as the time step size decreases.

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Numerical Simulation of Proppant Transport in Hydraulically Fractured Reservoirs

10.2118/203927-ms ◽

2021 ◽

Author(s):

Seyhan Emre Gorucu ◽

Vijay Shrivastava ◽

Long X. Nghiem

Keyword(s):

Hydraulic Fracturing ◽

Hydraulic Fracture ◽

Fractured Reservoirs ◽

Injection Timing ◽

Fracture Permeability ◽

Step Size ◽

Time Step ◽

Time Step Size ◽

Proppant Transport ◽

Fracture Closure

Abstract An existing equation-of-state compositional simulator is extended to include proppant transport. The simulator determines the final location of the proppant after fracture closure, which allows the computation of the permeability along the hydraulic fracture. The simulation then continues until the end of the production. During hydraulic fracturing, proppant is injected in the reservoir along with water and additives like polymers. Hydraulic fracture gets created due to change in stress caused by the high injection pressure. Once the fracture opens, the bulk slurry moves along the hydraulic fracture. Proppant moves at a different speed than the bulk slurry and sinks down by gravity. While the proppant flows along the fracture, some of the slurry leaks off into the matrix. As the fracture closes after injection stops, the proppant becomes immobile. The immobilized proppant prevents the fracture from closing and thus keeps the permeability of the fracture high. All the above phenomena are modelled effectively in this new implementation. Coupled geomechanics simulation is used to model opening and closure of the fracture following geomechanics criteria. Proppant retardation, gravitational settling and fluid leak-off are modeled with the appropriate equations. The propped fracture permeability is a function of the concentration of immobilized proppant. The developed proppant simulation feature is computationally stable and efficient. The time step size during the settling adapts to the settling velocity of the proppants. It is found that the final location of the proppants is highly dependent on its volumetric concentration and slurry viscosity due to retardation and settling effects. As the location and the concentration of the proppants determine the final fracture permeability, the additional feature is expected to correctly identify the stimulated region. In this paper, the theory and the model formulation are presented along with a few key examples. The simulation can be used to design and optimize the amount of proppant and additives, injection timing, pressure, and well parameters required for successful hydraulic fracturing.

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Studying Inertia Effects in Open Channel Flow Using Saint-Venant Equations

Water ◽

10.3390/w10111652 ◽

2018 ◽

Vol 10 (11) ◽

pp. 1652

Author(s):

Dong-Sin Shih ◽

Gour-Tsyh Yeh

Keyword(s):

Stokes Equations ◽

Field Application ◽

Benchmark Problems ◽

Algebraic Equations ◽

Step Size ◽

Time Step ◽

Cross Sectional ◽

Inertia Effects ◽

Time Step Size ◽

Saint Venant Equations

One-dimensional (1D) Saint-Venant equations, which originated from the Navier–Stokes equations, are usually applied to express the transient stream flow. The governing equation is based on the mass continuity and momentum equivalence. Its momentum equation, partially comprising the inertia, pressure, gravity, and friction-induced momentum loss terms, can be expressed as kinematic wave (KIW), diffusion wave (DIW), and fully dynamic wave (DYW) flow. In this study, the method of characteristics (MOCs) is used for solving the diagonalized Saint-Venant equations. A computer model, CAMP1DF, including KIW, DIW, and DYW approximations, is developed. Benchmark problems from MacDonald et al. (1997) are examined to study the accuracy of the CAMP1DF model. The simulations revealed that CAMP1DF can simulate almost identical results that are valid for various fluvial conditions. The proposed scheme that not only allows a large time step size but also solves half of the simultaneous algebraic equations. Simulations of accuracy and efficiency are both improved. Based on the physical relevance, the simulations clearly showed that the DYW approximation has the best performance, whereas the KIW approximation results in the largest errors. Moreover, the field non-prismatic case of the Zhuoshui River in central Taiwan is studied. The simulations indicate that the DYW approach does not ensure achievement of a better simulation result than the other two approximations. The investigated cross-sectional geometries play an important role in stream routing. Because of the consideration of the acceleration terms, the simulated hydrograph of a DYW reveals more physical characteristics, particularly regarding the raising and recession of limbs. Note that the KIW does not require assignment of a downstream boundary condition, making it more convenient for field application.

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